Real-rooted atlas

basis-map coefficients of A112227 → basis A013609 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 9 + 16\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1916429
22798608193
381544616192161449
42432882540029605603212077
572914896435643744012080153664110309
62187759383347404240882077604412840321281093001
765613840642495056448627231335369891841487361024025611653905
819683193344218233520453665604344144019126464420940847155225344512132827925

All 15 rows

Technical details
Node id
n-b35e74b1fec1401c · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      9,       16,         4;
   2:     27,       98,        60,          8;
   3:     81,      544,       616,        192,         16;
   4:    243,     2882,      5400,       2960,        560,         32;
   5:    729,    14896,     43564,      37440,      12080,       1536,         64;
   6:   2187,    75938,    334740,     424088,     207760,      44128,       4032,       128;
   7:   6561,   384064,   2495056,    4486272,    3133536,     989184,     148736,     10240,      256;
   8:  19683,  1933442,  18233520,   45366560,   43441440,   19126464,    4209408,    471552,    25344,     512;
   9:  59049,  9706576, 131501524,  444766080,  568588960,  335526912,  101394048,  16435200,  1424640,   61440,   1024;
  10: 177147, 48650978, 939923820, 4265897768, 7144010720, 5499027776, 2191964544, 482186496, 59938560, 4139520, 146432, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 16*x + 4*x^2.
%t Mathematica
P[1, x_] := 9 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9 + 16*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = (3 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.